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Download Poynting Vector and Power Flow in Electromagnetic Fields
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Poynting Vector and Power Flow in Electromagnetic Fields: Electromagnetic waves can transport energy from one point to another point. The electric and magnetic field intensities asscociated with a travelling electromagnetic wave can be related to the rate of such energy transfer. Let us consider Maxwell's Curl Equations: Using vector identity the above curl equations we can write .............................................(6.35) In simple medium where and are constant, we can write and Applying Divergence theorem we can write, ...........................(6.36) The term represents the rate of change of energy stored in the electric and magnetic fields and the term represents the power dissipation within the volume. Hence right hand side of the equation (6.36) represents the total decrease in power within the volume under consideration. The left hand side of equation (6.36) can be written as where 2 (W/mt ) is called the Poynting vector and it represents the power density vector associated with the electromagnetic field. The integration of the Poynting vector over any closed surface gives the net power flowing out of the surface. Equation (6.36) is referred to as Poynting theorem and it states that the net power flowing out of a given volume is equal to the time rate of decrease in the energy stored within the volume minus the conduction losses. Poynting vector for the time harmonic case: For time harmonic case, the time variation is of the form , and we have seen that instantaneous value of a quantity is the real part of the product of a phasor quantity and when is used as reference. For example, if we consider the phasor then we can write the instanteneous field as .................................(6.37) when E0 is Let us consider two instanteneous quantities A and B such that ..................(6.38) where A and B are the phasor quantities. real. i.e, Therefore, ..............................(6.39) Since A and B are periodic with period denoted by , the time average value of the product form AB, can be written as .....................................(6.40) Further, considering the phasor quantities A and B, we find that and , where * denotes complex conjugate. ..............................................(6.41) The poynting vector can be expressed as ...................................(6.42) If we consider a plane electromagnetic wave propagating in +z direction and has only component, from (6.42) we can write: Using (6.41) ........................................(6.43) where and , for the plane wave under consideration. For a general case, we can write .....................(6.44) We can define a complex Poynting vector and time average of the instantaneous Poynting vector is given by .